# Contents

## Idea

Extended quantum field theory (or multi-tiered quantum field theory) is the fully local formulation of functorial quantum field theory, formulated in higher category theory

Whereas a

• 1-categorical TQFT may be regarded as a rule that allows one to compute topological invariants $Z(\Sigma)$ assigned to $d$-dimensional manifolds by cutting these manifolds into a sequence $\{\Sigma_i\}$ of $d$-dimensional composable cobordisms with $(d-1)$-dimensional boundaries $\partial \Sigma_i$, e.g. $\Sigma = \Sigma_2 \coprod_{\partial \Sigma_1 = \partial \Sigma_2} \Sigma_1$, then assigning quantities $Z(\Sigma_i)$ to each of these and then composing these quantities in some way, e.g. as $Z(\Sigma) = Z(\Sigma_2)\circ Z(\Sigma_1)$;

we have that

• in extended TQFT $Z(\Sigma)$ may be computed by decomposing $\Sigma$ into $d$-dimensional pieces with piecewise smooth boundaries, whose boundary strata are of arbitrary codimension $k$.

For that reason extended QFT is also sometimes called local or localized QFT. In fact, the notion of locality in quantum field theory is precisely this notion of locality. And, as also discussed at FQFT, this higher dimensional version of locality is naturally encoded in terms of n-functoriality of $Z$ regarded as a functor on a higher category of cobordisms.

## Definition

### The category of extended cobordisms

The definition of a $j$-cobordism is recursive. A $(j+1)$-cobordism between $j$-cobordisms is a compact oriented $(j+1)$-dimensional smooth manifold with corners whose the boundary is the disjoint union of the target $j$-cobordism and the orientation reversal of the source $j$-cobordism. (The base case of the recursion is the empty set, thought of as a $(-1)$-dimensional manifold.)

$n Cob_d$ is an $n$-category with smooth compact oriented $(d-n)$-manifolds as objects and cobordisms of cobordisms up to $n$-cobordisms, up to diffeomorphism, as morphisms.

There are various suggestions with more or less detail for a precise definition of a higher category $n Cob_n$ of fully extended $n$-dimensional cobordisms.

A very general (and very natural) one consists in taking a further step in categorification: one takes $n$-cobordisms as $n$-morphisms and smooth homotopy classes of diffeomorphisms beween them as $(n+1)$-morphisms. Next one iterates this; see details at (∞,n)-category of cobordisms.

See

### Extended TQFT

Fix a base ring $R$, and let $C$ be the symmetric monoidal $n$-category of $n$-$R$-modules.

An $n$-extended $C$-valued TQFT of dimension $d$ is a symmetric $n$-tensor functor $Z: n Cob_d \rightarrow C$ that maps

• smooth compact oriented $d$-manifolds to elements of $R$

• smooth compact oriented $(d-1)$-manifolds to $R$-modules

• cobordisms of smooth compact oriented $(d-1)$-manifolds to $R$-linear maps between $R$-modules

• smooth compact oriented $(d-2)$-manifolds to $R$-linear additive categories

• cobordisms of smooth compact oriented $(d-2)$-manifolds to functors between $R$-linear categories

• etc …

• smooth compact oriented $(d-n)$-manifolds to $R$-linear $(n-1)$-categories

• cobordisms of smooth compact oriented $(d-n)$-manifolds to $(n-1)$-functors between $R$-linear $(n-1)$-categories

with compatibility conditions and gluing formulas that must be satisfied… For instance, since the functor $Z$ is required to be monoidal, it sends monoidal units to monoidal units. Therefore, the $d$-dimensional vacuum is mapped to the unit element of $R$, the $(d-1)$-dimensional vacuum to the $R$-module $R$, the $(d-2)$-dimensional vacuum to the category of $R$-modules, etc.

Here $n$ can range between $0$ and $d$. This generalizes to an arbitrary symmetric monoidal category $C$ as codomain category.

## Examples

### Classes of examples by dimension

$n=1$ gives ordinary TQFT.

The most common case is when $R = \mathbb{C}$ (the complex numbers), giving unitary ETQFT.

The most common cases for $C$ are

• $C = n Hilb(R)$, the category of $n$-Hilbert spaces? over a topological field $R$. As far as we know this is only defined up to $n=2$.

• $C = n Vect(R)$, the category of $n$-vector spaces over a field $R$.

• $C = n Mod(R)$, the (conjectured?) category of $n$-modules over a commutative ring $R$.

### Generic examples

By the cobordism hypothesis-theorem every fully dualizable object in a symmetric monoidal $(\infty,n)$-category with duals provides an example.

## Properties

### Construction of ETQFT’s

• By generators and relations

• By path integrals (this is Daniel Freed’s approach)

• By modular tensor n-categories?

### Classification of ETQFT’s

Assume $Z: n Cob_d \rightarrow n Vect(R)$ is an extended TQFT. Since $Z$ maps the $(d-1)$-dimensional vacuum to $R$ as an $R$-vector space, by functoriality $Z$ is forced to map a $d$-dimensional closed manifold to an element of $R$. Iterating this argument, one is naturally led to conjecture that, under the correct categorical hypothesis, the behaviour of $Z$ is enterely determined by its behaviour on $(d-n)$-dimensional manifolds. See details at cobordism hypothesis.

### Relation of ETQFT to AQFT

See

also

More on extended QFTs is also at

duality between algebra and geometry in physics:

## References

• Dan Freed, Remarks on Chern-Simons theory

• Daniel Freed, Quantum Groups from Path Integrals. arXiv

• Daniel Freed, Higher Algebraic Structures and Quantization. arXiv

• John Baez and James Dolan, Higher-dimensional Algebra and Topological Quantum Field Theory. arXiv

With an eye towards the full extension of Chern-Simons theory:

For TQFTs appearing in solid state physics in the context of topological order:

Revised on July 3, 2016 02:11:09 by Tim Porter (95.148.217.16)